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Equations, They ±ney describe aescriDe the me behavior Denavior of 01 the i derivatives of solutions with respect to initial conditions. rith. respect to initial conditions. 34) note by solu 0 with the of a fixed tie initial initial condition condition XQ, XQ, which which belongs belongs to to aa £-neighborhood £-nei point XQ. Let r* be the times at which the integral curve of the solution IQ. Let T* be the times at which the integral x =

19) t0 z(*o>£o) = £o, for t > to, can be written as x(t,x0) = X(t,t0)x0+ I X(t,T)f(r)dT+ ]T A"(*,r,- + 0)oi. 12) can be found in quadratures. CHAPTER 2. 1) the matrices A(t), Bi be constant. Ti, Ax\t=Tl = Bx. 22) Suppose that the times r; are indexed by the set of natural numbers such that Ti — ► +oo for i — ► oo. Without loss of generality, we can assume that Tl > t0. 23) where X(t,tQ) = eA^-T^ Yl (E + B)eA^~T^\ rQ = t0.

66) where v = y/g2 - A2, k 6 Z. -(-5+2k*). 67) Let us look at the motion that starts at the point 0, yo), where y0 = C0e-$Z. 69) and after that it will jump to a point of this line with y-coordinate yf — y\ + Io(yi)- After having made one more turn on the spiral, the phase point will get onto the semi-line x = 0, y > 0 with y-coordinate 2/2 = j/i e " , and after that, it will jump to the point of this semi-line with j/-coordinate equal to 2/^ = 2/2 + ^0(2/2) a Q d so on. Denote by h the mapping of the semi-line y > 0 into itself given by h:R+^R+, where A = ^2TT.